Dependence of the vertical magnetic field on intensity and umbral area. Top panel: vertical magnetic field averaged at fixed values of the continuum intensity plotted as a function of the area of the umbra for the individual spots. We consider four different values of the continuum intensity, 0.30 (blue), 0.475 (red), 0.60 (black), and 0.80 (green). The error bars correspond to the standard deviation of the magnetic field within the individual spots. The solid lines are linear fits for the different values of the continuum intensity. Bottom panel: estimate of the strength of B_z of the spines at I_C = 0.475 (blue squares, see text for more details) and the skewness of the histograms of B_z (red circles) at this intensity as a function of the umbral area. The horizontal line indicates the mean B_z at I_C = 0.475 averaged over all spots in our sample.